Method for designing freeform surface

ABSTRACT

A method for designing freeform surface is provided. An initial surface is established. A plurality of feature rays are selected. A plurality of intersections of the plurality of feature rays with an unknown freeform surface are calculated based on a given object-image relationship and a vector form of the Snell&#39;s law. The plurality of intersections are a plurality of feature data points. An unknown freeform surface equation is obtained by surface fitting the plurality of feature data points.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims all benefits accruing under 35 U.S.C. §119 fromChina Patent Application No. 201410077829.1, field on Mar. 5, 2014 inthe China Intellectual Property Office, disclosure of which isincorporated herein by reference. The application is also related tocopending applications entitled, “OFF-AXIAL THREE-MIRROR OPTICAL SYSTEMWITH FREEFORM SURFACES”, filed ______ (Atty. Docket No. US55059);“OFF-AXIAL THREE-MIRROR OPTICAL SYSTEM WITH FREEFORM SURFACES”, filed______ (Atty. Docket No. US55058); “METHOD FOR DESIGNING OFF-AXIALTHREE-MIRROR OPTICAL SYSTEM WITH FREEFORM SURFACES”, filed ______ (Atty.Docket No. US55960).

BACKGROUND

1. Technical Field

The present disclosure relates to a method for designing freeformsurface, especially a method based on a point-by-point construction.

2. Description of Related Art

Compared with conventional rotationally symmetric surfaces, freeformoptical surfaces have higher degrees of freedom, which can reduce theaberrations and simplify the structure of the system in optical design.In recent years, with the development of the advancing manufacturetechnologies, freeform surfaces have been successfully used in theoptical field, such as head-mounted-displays, reflective systems,varifocal panoramic optical systems, and micro-lens arrays.

However, conventional methods mostly focus on designing atwo-dimensional contour for freeform surfaces, and they are applied onlyto design optical systems with small aperture and linear field-of-view.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the embodiments can be better understood with referenceto the following drawings. The components in the drawings are notnecessarily drawn to scale, the emphasis instead being placed uponclearly illustrating the principles of the embodiments. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 shows a flow chart of one embodiment of a method for designingfreeform surface.

FIG. 2 is a schematic view of a selecting method of a plurality offeature rays employed in each field.

FIG. 3 is a schematic view of start point and end point of one featureray while solving the feature data points.

FIG. 4 shows a schematic view of one embodiment of a freeform surfaceobtained by the method in FIG. 1.

DETAILED DESCRIPTION

It will be appreciated that for simplicity and clarity of illustration,where appropriate, reference numerals have been repeated among thedifferent figures to indicate corresponding or analogous elements. Inaddition, numerous specific details are set forth in order to provide athorough understanding of the embodiments described herein. However, itwill be understood by those of ordinary skill in the art that theembodiments described herein can be practiced without these specificdetails. In other instances, methods, procedures and components have notbeen described in detail so as not to obscure the related relevantfeature being described. The drawings are not necessarily to scale andthe proportions of certain parts may be exaggerated to better illustratedetails and features. The description is not to be considered aslimiting the scope of the embodiments described herein.

Several definitions that apply throughout this disclosure will now bepresented.

The term “comprising” means “including, but not necessarily limited to”;it specifically indicates open-ended inclusion or membership in aso-described combination, group, series, and the like. The term“substantially” is defined to be essentially conforming to theparticular dimension, shape or other word that substantially modifies,such that the component need not be exact. For example, substantiallycylindrical means that the object resembles a cylinder, but can have oneor more deviations from a true cylinder. It should be noted thatreferences to “one” embodiment in this disclosure are not necessarily tothe same embodiment, and such references mean at least one.

Referring to FIG. 1, a method for designing freeform surface of oneembodiment is provided. The method for designing freeform surfaceincludes the following steps:

step (S1), establishing an initial surface;

step (S2), selecting a plurality of feature rays R_(i) (i=1, 2 . . . K);

step (S3), calculating a plurality of intersections of the plurality offeature rays R_(i) (i=1, 2 . . . K) with an unknown freeform surfacebased on a given object-image relationship and Snell's law, wherein theplurality of intersections are a plurality of feature data points P_(i)(i=1, 2 . . . K); and

step (S4), obtaining an unknown freeform surface equation by surfacefitting the plurality of feature data points P_(i) (i=1, 2 . . . K).

In step (S1), the initial surface can be planar, spherical, or othersurface types. An initial surface location can be selected according tothe optical system actual needs. In one embodiment, the initial surfaceis a planar.

In step (S2), a plurality of intersections of the plurality of featurerays R_(i) (i=1, 2 . . . K) and an image surface are close to aplurality of ideal image points I_(i) (i=1, 2 . . . K). The selecting aplurality of feature rays R_(i) (i=1, 2 . . . K) comprises includessteps of: M fields are selected according to the optical systems actualneeds; an aperture of each of the M fields is divided into N equalparts; and, P feature rays at different aperture positions in each ofthe N equal parts are selected. As such, K=M×N×P different feature rayscorresponding to different aperture positions and different fields arefixed. The aperture can be circle, rectangle, square, oval or othershapes.

FIG. 2 illustrates that in one embodiment, the aperture of each of the Mfields is a circle, and a circular aperture of each of the M fields isdivided into N angles with equal interval φ, as such, N=2π/φ; then, Pdifferent aperture positions are fixed along a radial direction of eachof the N angles. Therefore, K=M×N×P different feature rays correspondingto different aperture positions and different fields are fixed. In oneembodiment, six fields are fixed in the construction process; a circularaperture of each of the six fields is divided into fourteen angles withequal intervals; and seven different aperture positions are fixed alongthe radial direction of each of the fourteen angles. Therefore, 588different feature rays corresponding to different aperture positions anddifferent fields are fixed.

In step (S3), referring to FIG. 3, a surface S2 is defined as theunknown freeform surface, a surface Ω′ is defined as a surface locatedadjacent to and before the surface Ω, and a surface Ω″ is defined as asurface located adjacent to and behind the surface Ω. Defining theintersections of the plurality of feature rays R_(i) (i=1, 2 . . . K)with the surface Ω as the feature data points P_(i) (i=1, 2 . . . K).The feature data points P_(i) (i=1, 2 . . . K) can be obtained by theintersections of the feature rays R_(i) (i=1, 2 . . . K) with thesurface Ω′ and the surface Ω″. The plurality of feature rays R_(i) (i=1,2 . . . K) are intersected with Ω′ at a plurality of start points S_(i)(i=1, 2 . . . K), and intersected with Ω″ at a plurality of end pointsE_(i) (i=1, 2 . . . K). When the surface Ω and the plurality of featurerays R_(i) (i=1, 2 . . . K) are determined, the plurality of startpoints S_(i) (i=1, 2 . . . K) of the feature rays R_(i) (i=1, 2 . . . K)can also be determined. The plurality of end points E_(i) (i=1, 2 . . .K) can be obtained based on the object-image relationship. Under idealconditions, the feature rays R_(i) (i=1, 2 . . . K) emit from theplurality of start points S_(i) (i=1, 2 . . . K) on Ω′; pass through thefeature data points P_(i) (i=1, 2 . . . K) on the surface Ω; intersectwith Ω″ at the plurality of end points E_(i) (i=1, 2 . . . K); andfinally intersect with the image plane at the plurality of ideal imagepoints I_(i) (i=1, 2 . . . K). If the surface Ω″ is the image plane, theplurality of end points E_(i) (i=1, 2 . . . K) are the plurality ofideal image points I_(i) (i=1, 2 . . . K). If there are other surfacesbetween the surface Ω and the image plane, the plurality of end pointsE_(i) (i=1, 2 . . . K) are the points on the surface Ω″ which minimizesan optical path length between the feature data points P_(i) (i=1, 2 . .. K) and the ideal image points I_(i) (i=1, 2 . . . K).

The plurality of feature data points P_(i) (i=1, 2 . . . K) can beobtained by the following two calculating methods.

A first calculating method includes the following sub-steps:

Step (S31): defining a first intersection of a first feature ray R₁ andthe initial surface as a feature data point P₁;

Step (S32): when i (1≦i≦K−1) feature data points P_(i) (1≦i≦K−1) havebeen obtained, a unit normal vector {right arrow over (N)}_(i) (1≦i≦K−1)at each of the i (1≦i≦K−1) feature data points P_(i) (1≦i≦K−1) can becalculated based on a vector form of Snell's Law;

Step (S33): making a first tangent plane at the i (1≦i≦K−1) feature datapoints P_(i) (1≦i≦K−1) respectively; thus i first tangent planes can beobtained, and i×(K−i) second intersections can be obtained by the ifirst tangent planes intersecting with remaining (K−i) feature rays; anda second intersection, which is nearest to the i (1≦i≦K−1) feature datapoints P_(i), is fixed from the i×(K−i) second intersections as a nextfeature data point P_(i+1) (1≦i≦K−1); and

Step (S34): repeating steps S32 and S33, until all the plurality offeature data points P_(i) (i=1, 2 . . . K) are calculated.

In step (S32), the unit normal vector {right arrow over (N)}_(i)(1≦i≦K−1) at each of the feature data point P_(i) (1≦i≦K−1) can becalculated based on the vector form of Snell's Law. When the unknownfreeform surface is a refractive second surface,

$\begin{matrix}{{\overset{\rightarrow}{N}}_{i} = \frac{{n^{\prime}{\overset{\rightarrow}{r}}_{i}^{\prime}} - {n{\overset{\rightarrow}{r}}_{i}}}{{{n^{\prime}{\overset{\rightarrow}{r}}_{i}^{\prime}} - {n{\overset{\rightarrow}{r}}_{i}}}}} & (1)\end{matrix}$

is a unit vector along a direction of an incident ray of the unknownfreeform surface;

${\overset{\rightarrow}{r}}_{i}^{\prime} = \frac{\overset{\rightharpoonup}{E_{i}P_{i}}}{\overset{\rightharpoonup}{E_{i}P_{i}}}$

is a unit vector along a direction of an exit ray of the unknownfreeform surface; and n, n′ is refractive index of a media at twoopposite sides of the unknown freeform surface respectively.

Similarly, when the unknown freeform surface is a reflective surface,

$\begin{matrix}{{\overset{\rightarrow}{N}}_{i} = \frac{{\overset{\rightarrow}{r}}_{i}^{\prime} - {\overset{\rightarrow}{r}}_{i}}{{{\overset{\rightarrow}{r}}_{i}^{\prime} - {\overset{\rightarrow}{r}}_{i}}}} & (2)\end{matrix}$

The unit normal vector {right arrow over (N)}_(i) at the feature datapoints P_(i) (i=1, 2 . . . K) is perpendicular to the first tangentplane at the feature data points P_(i) (i=1, 2 . . . K). Thus, the firsttangent planes at the feature data points P_(i) (i=1, 2 . . . K) can beobtained.

The first calculating method includes a computational complexity formulaof

${T(K)} = {{\sum\limits_{i = 1}^{K - 1}{i\left( {K - i} \right)}} = {{{\frac{1}{6}K^{3}} - {\frac{1}{6}K}} = {{O\left( K^{3} \right)}.}}}$

When multi-feature rays are used in a design, the first calculatingmethod requires a long computation time.

A second calculating method includes the following sub-steps:

Step (S′31): defining a first intersection of a first feature light rayR₁ and the initial surface as a feature data point P₁;

Step (S′32): when an ith (1≦i≦K−1) feature data point P_(i) (1≦i≦K−1)has been obtained, a unit normal vector {right arrow over (N)}_(i) atthe ith (1≦i≦K−1) feature data point P_(i) (1≦i≦K−1) can be calculatedbased on the vector form of Snell's law;

Step (S′33): making a first tangent plane through the ith (1≦i≦K−1)feature data point P_(i) (1≦i≦K−1), and (K−i) second intersections canbe obtained by the first tangent plane intersecting with remaining (K−i)feature rays; a second intersection Q_(i+1), which is nearest to the ith(1≦i≦K−1) feature data point P_(i) (1≦i≦K−1), is fixed; and a featureray corresponding to the second intersection Q_(i+1) is defined asR_(i+1), a shortest distance between the second intersection Q_(i+1) andthe ith feature data point P_(i) (1≦i≦K−1) is defined as d_(i);

Step (S′34): making a second tangent plane at (i−1) feature data pointsthat are obtained before the ith feature data point P_(i) (1≦i≦K−1)respectively; thus, (i−1) second tangent planes can be obtained, and(i−1) third intersections can be obtained by the (i−1) second tangentplanes intersecting with a feature ray R_(i+1); in each of the (i−1)second tangent planes, each of the third intersections and itscorresponding feature data point form an intersection pair; theintersection pair, which has the shortest distance between a thirdintersection and its corresponding feature data point, is fixed; and thethird intersection and the shortest distance is defined as Q′_(i+1) andd′_(i) respectively;

Step (S′35): comparing d_(i) and d′_(i), if d_(i)≦d′_(i), Q_(i+1) istaken as the next feature data point P_(i+1) (1≦i≦K−1); otherwise,Q′_(i+1) is taken as the next feature data point P_(i+1) (1≦i≦K−1); and

Step (S′36): repeating steps from S′32 to S′35, until the plurality offeature data points P_(i) (i=1, 2 . . . K) are all calculated.

In Step (S′32), a calculating method of the unit normal vector {rightarrow over (N)}_(i) at the ith (1≦i≦K−1) feature data point P_(i)(1≦i≦K−1) is the same as the first calculating method.

A second calculating method includes a computational complexity formulaof

${T(K)} = {{{\sum\limits_{i = 1}^{K - 1}K} - i + i - 1} = {\left( {K - 1} \right)^{2} = {{O\left( K^{2} \right)}.}}}$

When multi-feature rays are used in a design, the computationalcomplexity of the second calculating method is smaller than thecomputational complexity of the first calculating method. In oneembodiment, constructing the plurality of feature data points P_(i)(i=1, 2 . . . K) point by point using the second calculating method.

In some embodiments, the freeform surface obtained in one embodiment canbe the initial surface for further optimization.

Referring to FIG. 4, in one embodiment, the plurality of feature datapoints P_(i) (i=1, 2 . . . 588) are obtained by the second calculatingmethod, the plurality of feature data points P_(i) (i=1, 2 . . . 588)are the intersections of the different feature rays and the unknownfreeform surface. The unknown freeform surface is obtained by surfacefitting the plurality of feature data points P_(i) (i=1, 2 . . . 588).

The designing method of freeform surface is based on a point by pointconstruction, and optical relationships of the feature rays formedbetween the unknown freeform surface and the surfaces adjacent to theunknown freeform surface, a three-dimensional freeform surface can beobtained by the designing method. And the designing method is simple andcan be applied to various off-axis asymmetric systems. Furthermore, thedesigning method can be applied to imaging systems with multi-fields andcertain aperture, by controlling the feature rays of the multi-fieldsand different aperture positions; and the number of fields is notlimited, thus, the design method has a broad application use.

It is to be understood that the above-described embodiments are intendedto illustrate rather than limit the disclosure. Any elements describedin accordance with any embodiments is understood that they can be usedin addition or substituted in other embodiments. Embodiments can also beused together. Variations may be made to the embodiments withoutdeparting from the spirit of the disclosure. The above-describedembodiments illustrate the scope of the disclosure but do not restrictthe scope of the disclosure.

Depending on the embodiment, certain of the steps of methods describedmay be removed, others may be added, and the sequence of steps may bealtered. It is also to be understood that the description and the claimsdrawn to a method may include some indication in reference to certainsteps. However, the indication used is only to be viewed foridentification purposes and not as a suggestion as to an order for thesteps.

What is claimed is:
 1. A method for designing freeform surfacecomprising: step (S1), establishing an initial surface; step (S2),selecting a plurality of feature rays R_(i) (i=1, 2 . . . K); step (S3),calculating a plurality of intersections of the plurality of featurerays R_(i) (i=1, 2 . . . K) with an unknown freeform surface based on agiven object-image relationship and a vector form of the Snell's law,wherein the plurality of intersections are a plurality of feature datapoints P_(i) (i=1, 2 . . . K); and step (S4), obtaining an unknownfreeform surface equation by surface fitting the plurality of featuredata points P_(i) (i=1, 2 . . . K).
 2. The method of claim 1, whereinthe initial surface is planar, curved, or spherical.
 3. The method ofclaim 1, wherein the selecting a plurality of feature rays R_(i) (i=1,
 2. . . K) comprises steps of: M fields are selected; an aperture of eachof the M fields is divided into N equal parts; and, P feature rays atdifferent positions in each of the N equal parts are selected, thus,K=M×N×P different feature rays are selected.
 4. The method of claim 3,wherein the aperture of each of the M fields is a circle.
 5. The methodof claim 4, wherein the aperture of each of the M fields is divided intoN angles with equal intervals.
 6. The method of claim 5, wherein Pdifferent positions are selected along a radial direction of each of theN angles.
 7. The method of claim 1, wherein Ω is defined as the unknownfreeform surface, Ω′ is defined as a surface located adjacent to andbefore Ω, and Ω″ is defined as a surface located adjacent to and behindΩ; and the plurality of feature rays R_(i) (i=1, 2 . . . K) and Ω′intersect at a plurality of start points S_(i) (i=1, 2 . . . K), and theplurality of feature rays R_(i) (i=1, 2 . . . K) and Ω″ intersect at aplurality of end points E_(i) (i=1, 2 . . . K).
 8. The method of claim7, wherein the plurality of end points E_(i) (i=1, 2 . . . K) areobtained by the given object-image relationship.
 9. The method of claim8, wherein Ω″ is the image plane, and the plurality of end points E_(i)(i=1, 2 . . . K) are the plurality of ideal image points I_(i) (i=1, 2 .. . K).
 10. The method of claim 7, wherein there are other surfacesbetween the unknown freeform surface and the image plane, and theplurality of end points E_(i) (i=1, 2 . . . K) are a plurality of pointson Ω″ which minimizes an optical path length between the plurality offeature data points P_(i) (i=1, 2 . . . K) and the plurality of idealimage points I_(i) (i=1, 2 . . . K).
 11. The method of claim 7, whereina unit normal vector {right arrow over (N)}_(i) at each of the pluralityof feature data points P_(i) (i=1, 2 . . . K) is calculated based on thevector form of Snell's law, the unknown freeform surface is a refractivesurface, and:${{\overset{\rightarrow}{N}}_{i} = \frac{{n^{\prime}{\overset{\rightarrow}{r}}_{i}^{\prime}} - {n{\overset{\rightarrow}{r}}_{i}}}{{{n^{\prime}{\overset{\rightarrow}{r}}_{i}^{\prime}} - {n{\overset{\rightarrow}{r}}_{i}}}}};$wherein${\overset{\rightarrow}{r}}_{i} = \frac{\overset{\rightharpoonup}{P_{i}S_{i}}}{\overset{\rightharpoonup}{P_{i}S_{i}}}$is a unit vector along a direction of an incident ray of the unknownfreeform surface;${\overset{\rightarrow}{r}}_{i}^{\prime} = \frac{\overset{\rightharpoonup}{E_{i}P_{i}}}{\overset{\rightharpoonup}{E_{i}P_{i}}}$is a unit vector along a direction of an exit ray of the unknownfreeform surface; and n, n′ is refractive index of a media at twoopposite sides of the unknown freeform surface respectively.
 12. Themethod of claim 7, wherein a unit normal vector {right arrow over(N)}_(i) at each of the feature data points P_(i) is calculated based onthe vector form of Snell's law, the unknown freeform surface is areflective surface, and:${{\overset{\rightarrow}{N}}_{i} = \frac{{\overset{\rightarrow}{r}}_{i}^{\prime} - {\overset{\rightarrow}{r}}_{i}}{{{\overset{\rightarrow}{r}}_{i}^{\prime} - {\overset{\rightarrow}{r}}_{i}}}};$wherein${\overset{\rightarrow}{r}}_{i} = \frac{\overset{\rightharpoonup}{P_{i}S_{i}}}{\overset{\rightharpoonup}{P_{i}S_{i}}}$is a unit vector along a direction of an incident ray of the unknownfreeform surface;${\overset{\rightarrow}{r}}_{i}^{\prime} = \frac{\overset{\rightharpoonup}{E_{i}P_{i}}}{\overset{\rightharpoonup}{E_{i}P_{i}}}$is a unit vector along a direction of an exit ray of the unknownfreeform surface.
 13. The method of claim 1, wherein the calculating aplurality of intersections of the plurality of feature rays R_(i) (i=1,2 . . . K) with an unknown freeform surface based on a givenobject-image relationship and a vector form of the Snell's lawcomprises: Step (S31): defining a first intersection of a first featureray R₁ and the initial surface as a feature data point P₁; Step (S32):when i (1≦i≦K−1) feature data points P_(i) (1≦i≦K−1) have been obtained,a unit normal vector {right arrow over (N)}_(i) (1≦i≦K−1) at each of thei (1≦i≦K−1) feature data points P_(i) (1≦i≦K−1) is calculated based on avector form of Snell's law; Step (S33): making a first tangent plane atthe i (1≦i≦K−1) feature data points P_(i) (1≦i≦K−1) respectively; thus ifirst tangent planes are obtained, and i×(K−i) second intersections areobtained by the i first tangent planes intersecting with remaining (K−i)feature rays; and a second intersection, which is nearest to the i(1≦i≦K−1) feature data points P_(i), is fixed from the i×(K−i) secondintersections as a next feature data point P_(i+1) (1≦i≦K−1); and Step(S34): repeating steps (S32) and (S33), until all the plurality offeature data points P_(i) (i=1, 2 . . . K) are calculated.
 14. Themethod of claim 13, wherein the calculating method includes acomputational complexity of${T(K)} = {{\sum\limits_{i = 1}^{K - 1}{i\left( {K - i} \right)}} = {{{\frac{1}{6}K^{3}} - {\frac{1}{6}K}} = {{O\left( K^{3} \right)}.}}}$15. The method of claim 1, wherein the calculating a plurality ofintersections of the plurality of feature rays R_(i) (i=1, 2 . . . K)with an unknown freeform surface based on a given object-imagerelationship and a vector form of the Snell's law comprises: Step(S′31): defining a first intersection of a first feature ray R₁ and theinitial surface as a feature data point P₁; Step (S′32): when an ith(1≦i≦K−1) feature data point P_(i) (1≦i≦K−1) has been obtained, a unitnormal vector {right arrow over (N)}_(i) at the ith (1≦i≦K−1) featuredata point P_(i) (1≦i≦K−1) is calculated based on the vector form ofSnell's law; Step (S′33): making a first tangent plane through the ith(1≦i≦K−1) feature data point P_(i) (1≦i≦K−1), and (K−i) secondintersections are obtained by the first tangent plane intersecting withremaining (K−i) feature rays; a second intersection Q_(i+1), which isnearest to the ith (1≦i≦K−1) feature data point P_(i) (1≦i≦K−1), isfixed; and a feature ray corresponding to the second intersectionQ_(i+1) is defined as R_(i+1), a shortest distance between the secondintersection Q_(i+1) and the ith (1≦i≦K−1) feature data point P_(i)(1≦i≦K−1) is defined as d_(i); Step (S′34): making a second tangentplane at (i−1) feature data points that are obtained before the ith(1≦i≦K−1) feature data point P_(i) (1≦i≦K−1) respectively; thus, (i−1)second tangent planes are obtained, and (i−1) third intersections areobtained by the (i−1) second tangent planes intersect with a feature rayR_(i+1); in each of the (i−1) second tangent planes, each of the thirdintersections and its corresponding feature data point form anintersection pair; the intersection pair, which has the shortestdistance between a third intersection and its corresponding feature datapoint, is fixed; and the third intersection and the shortest distance isdefined as Q′_(i+1) and d′_(i) respectively; Step (S′35): comparingd_(i) and d′_(i); if d_(i)≦d′_(i), Q_(i+1) is taken as a next featuredata point P_(i+1) (1≦i≦K−1); otherwise, Q′_(i+1) is taken as the nextfeature data point P_(i+1) (1≦i≦K−1); and Step (S′36): repeating stepsfrom (S′32) to (S′35), until all the plurality of feature data pointsP_(i) (i=1, 2 . . . K) are calculated.
 16. The method of claim 15,wherein the calculating method includes a computational complexity of${T(K)} = {{{\sum\limits_{i = 1}^{K - 1}K} - i + i - 1} = {\left( {K - 1} \right)^{2} = {{O\left( K^{2} \right)}.}}}$